C++ Binary Fixed-Point Arithmetic

ISO/IEC JTC1 SC22 WG21 N3352 = 12-0042 - 2012-01-15

Lawrence Crowl, Lawrence@Crowl.org


C++ supports integer arithmetic and floating-point arithmetic, but it does not support fixed-point arithmetic. We propose support for fixed-point arithmetic via standard library facilities.

Fixed-Point versus Integer

In C and C++, the basic integer types have several problematic behaviors.

Fixed-Point versus Floating-Point

Fixed-point arithmetic is better than floating-point arithmetic in several domains.

Prior Art

The popular computing literature abounds with articles on how to use integers to implement fixed-point arithmetic. However, manually writing fixed-point arithmetic with integer arithmetic is tedious and prone to error. Direct support is desirable.

ISO/IEC TR 18037 [E] provides fixed-point support in the C programming language. However, this support is not general; only a few possible radix positions are supported. The feature is essentially limited to the digital signal processing domain.

Likewise, software implementations of fixed-point arithmetic in C and C++, e.g. libfixmath [F], are also not general as the support only a limited number of radix positions.

The programming languages Ada [A], COBOL [B], CORAL 66 [C1] [C2], JOVIAL [J], and PL/I [P] provide direct support for fixed-point arithmetic.

Proposal Outline

We propose extending the standard library to provide general purpose binary fixed-point arithmetic. We anticipate that these library components will be used for the manipulation of program data variables, and not program control variables. That is, we do not view the proposal as a replacement for index or size variables.

The design relies on generally 'safe' defaults, but with additional explicit controls to match particular application domains or to enable more efficient execution.

Our proposal requires no new hardware, and is implementable as a pure library. Indeed, much of that implementation already exists. However, some operations could be substantially faster with direct hardware support.

Basic Types

The fixed-point library contains four class templates. They are cardinal and integral for integer arithmetic, and nonnegative and negatable for fractional arithmetic.

These types have a range specified by an integer. The range of an unsigned number n is 0 <= n < 2g where g is the range parameter. The range of an signed number n is 2g < n < 2g. Note that the range interval is half-open for unsigned numbers and open for signed numbers. For example, cardinal<8> has values n such that 0 <= n < 256 and integral<8> has values n such that -256 < n < 256.

The fractional types have a resolution specified by an integer. The resolution of a fractional number n is 2s, where s is the resolution parameter. For example, negatable<8,-4> has values n such that -256 < n < 256 in increments of 2-4 = 1/16.

Both range and resolution parameters may be either positive or negative. The number of significant bits is g-s. This specification enables representing both very small and very large values with few bits. In any event, the range must be greater than the resolution, that is g>s.

Basic Operations

The basic arithmetic operations are addition, subtraction, multiplication and division. When mixing operands of different template class types, cardinal types will promote to the other types, and the other types will promote to negatable type. The effect is that unsigned fixed-point types promote to signed fixed-point types. There are notable exceptions. Negation and subtraction on unsigned types yields a signed type. Comparison across types is direct; there is no conversion beforehand.

In general, the range and resolution of the result of basic operations are large enough to hold the mathematical results. The following table shows the range and resolution for the results of basic operations on fractional types. The $ operator returns the range parameter. The @ operator returns the resolution parameter.

operationresult rangeresult resolution

Overflow in template argument computation is undefined behavior. In practice, overflow is unlikely to be a significant problem because even small machines can represent numbers with thousands of bits and because compiler can diagnose overflow in template arguments.

The special case in the operations above is division, where the mathematical result may require an infinite number of bits. The actual value must be rounded to a representable value. The above resolution is sufficient to ensure that if the mathematical result is not zero, the fixed-point result is not zero. Furthermore, assuming values have an error of one-half ULP, the defined resolution is close to the error bound in the computation.

When the computation is not exact, rounding will be to one of the two nearest representable values. The algorithm for choosing between these values is the rounding mode. Different applications desire different modes, so programmers may specify the rounding mode with a value of type enum class round. The possible values are:

Speed is more important than the choice in value.
Round towards negative infinity. This mode is useful in interval arithmetic.
Round towards zero. This mode is useful in implementing integral arithmetic.
Round towards positive infinity. This mode is useful in interval arithmetic.
Round towards the nearest value, but exactly-half values are rounded towards maximum magnitude. This mode is the standard school algorithm.
Round towards the nearest value, but exactly-half values are rounded towards even values. This mode has more balance than the classic mode.
Round towards the nearest value, but exactly-half values are rounded towards odd values. This mode has as much balance as the near_even mode, but preserves more information.

In general, these modes get slower but more accurate working down the list.

Construction and Assignment

Since the range of intermediate values grow to hold all possible values, and variables have a static range and resolution, construction and assignment may need to reduce the range and resolution. Reducing the resolution is done with a rounding mode associated with the variable. When the dynamic value exceeds the range of variable, the assignment overflows.

When an overflow does occur, the desirable behavior depends on the application, so programmers may specify the overflow mode with a value of type enum class overflow. The possible values are:

Programmer analysis of the program has determined that overflow cannot occur. Uses of this mode should be accompanied by an argument supporting the conclusion.
Programmers are willing to accept undefined behavior in the event of an overflow.
The assigned value is the dynamic value mod the range of the variable. This mode makes sense only with unsigned numbers. It is useful for angular measures.
If the dynamic value exceeds the range of the variable, assign the nearest representable value.
If the dynamic value exceeds the range of the variable, throw an exeception of type std::overflow_error.

In general, these modes get slower but safer working down the list.


There exists no mechanism in C++11 to specify literals for the template types above. However, we can get close with template functions that yield the appropriate fixed-point value based on an template int parameter. For example, the expression to_cardinal<24>() will produce a cardinal constant with a range just sufficient to hold the value 24. Likewise, the expression to_nonnegative<2884,-4>() will produce a nonnegative constant with a range and resolution just sufficient to hold the value 2884*2-4.


Alpha Blending

Consider the alpha blending of two RGBA pixels, a and b. To avoid redundancy, we will only show computation for the red color. The algorithm is somewhat complicated by the need to convert the [0-255] range of color representation to the [0-1] range of color values.

struct pixel { cardinal<8> r, g, b, a; };

pixel blend( pixel a, pixel b ) {
  constexpr scale = to_nonnegative<255,0>;
  auto a_r = a.r / scale;
  auto b_r = b.r / scale;
  auto aia = b.a * (to_cardinal<1>() - a.a);
  auto c_a = a_a + aia;
  auto c_r = (a.r*a.a + b.r*aia) / c_a;
  pixel c;
  c.a = static_cast<nonnegative<8,0>(c_a * to_nonnegative<255,0>);
  c.r = static_cast<nonnegative<8,0>(c_r * to_nonnegative<255,0>);
  return c;

Proposal Details

Type Signatures

The template class type signatures are as follows.

    int Crng,
    overflow Covf = overflow::exception
class cardinal;
    int Crng,
    overflow Covf = overflow::exception
class integral;
    int Crng,
    int Crsl,
    round Crnd = round::nearest_odd,
    overflow Covf = overflow::exception
class nonnegative;
    int Crng,
    int Crsl,
    round Crnd = round::nearest_odd,
    overflow Covf = overflow::exception
class negatable;


operations types notes
default construction all uninitialized
copy construction all identical value
value construction from same or lower type value subject to overflow and/or rounding
v++; v--; ++v; --v
cardinal and integral value subject to overflow
-v all result is a signed type
!v all test for value != 0
~v cardinal invert bits, but still within range
v.scale_up<n>() all multiply by 2n, where n>0
v.scale<n,r>() all multiply by 2n, apply the rounding mode when n<0
v*u all multiplication
v.divide<round>(u) cardinal and integral integer division with the given roundng mode
v.divide<round>(u) nonnegative and negatable fractional division with the given roundng mode
v/u cardinal and integral v.divide<truncated>(u)
v/u nonnegative and negatable v.divide<class-specified rounding mode>(u)
v/u nonnegative and negatable v.divide<class-specified rounding mode>(u)
v%u cardinal and integral remainder
v+u all addition
v-u all subtraction, the result is always signed
comparisons all no value promotion/conversion, all comparisons are value-based
v&u; v^u; v|u cardinal bitwise logical operations
v=u; a*=b; a/=b; a%=b; a+=b; a-=b; all (compound) assignment
v&=u; v^=u; v|=u; cardinal bitwise logical compound assignment


Ada Reference Manual, http://www.ada-auth.org/arm.html
ISO IEC JTC1/SC22/WG4 - COBOL, http://www.cobolstandard.info/wg4/wg4.html
BS 5905:1980 Specification for computer programming language CORAL 66, October 1980, http://shop.bsigroup.com/ProductDetail/?pid=000000000000133933
XGC CORAL 66 Language Reference Manual, 2001, http://www.xgc.com/manuals/xgc-c66-rm/book1.html
ISO/IEC TR 18037:2008: Programming languages -- C -- Extensions to support embedded processors, http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=51126
libfixmath: Cross Platform Fixed Point Maths Library, http://code.google.com/p/libfixmath
MIL-STD-1589C (USAF) JOVIAL (J73), 6 July 1984, http://www.everyspec.com/MIL-STD/MIL-STD+%281500+-+1599%29/MIL-STD-1589C_14577/
PL/I, Wikipedia, http://en.wikipedia.org/wiki/PL/I